Theorem epse 4097

Description: The epsilon relation is set-like on any class. (This is the origin of the term "set-like": a set-like relation "acts like" the epsilon relation of sets and their elements.) (Contributed by Mario Carneiro, 22-Jun-2015.)

Assertion

Ref	Expression
epse	⊢ E Se 𝐴

Proof of Theorem epse

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		epel 4047	. . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
2	1	bicomi 130	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ 𝑦 E 𝑥)
3	2	abbi2i 2193	. . . . 5 ⊢ 𝑥 = {𝑦 ∣ 𝑦 E 𝑥}
4		vex 2604	. . . . 5 ⊢ 𝑥 ∈ V
5	3, 4	eqeltrri 2152	. . . 4 ⊢ {𝑦 ∣ 𝑦 E 𝑥} ∈ V
6		rabssab 3081	. . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ⊆ {𝑦 ∣ 𝑦 E 𝑥}
7	5, 6	ssexi 3916	. . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V
8	7	rgenw 2418	. 2 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V
9		df-se 4088	. 2 ⊢ ( E Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦 E 𝑥} ∈ V)
10	8, 9	mpbir 144	1 ⊢ E Se 𝐴

Syntax hints:   ∈ wcel 1433  {cab 2067  ∀wral 2348  {crab 2352  Vcvv 2601   class class class wbr 3785   E cep 4042   Se wse 4084

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-eprel 4044  df-se 4088

This theorem is referenced by: (None)